The system of equations $4x + y + 2z = 5$, $x - 5y + 3z = 10$, and $9x - 3y + 7z = 20$ has

If the system of linear equations $2x + 2y + 3z = a$,$3x - y + 5z = b$,and $x - 3y + 2z = c$,where $a, b, c$ are non-zero real numbers,has more than one solution,then:

If the system of linear equations: $x + y + z = 6, x + 2y + 5z = 10, 2x + 3y + \lambda z = \mu$ has infinitely many solutions, then the value of $\lambda + \mu$ equals:

The system of linear equations $(\sin \theta) x + y - 2z = 0$, $2x - y + (\cos \theta) z = 0$, and $-3x + (\sec \theta) y + 3z = 0$, where $\theta \neq (2n + 1) \frac{\pi}{2}$, has a non-trivial solution for:

Let $A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 1 & 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 2 \\ 1 \\ 7 \end{bmatrix}$. For the equation $AX = B$, find the matrix $X$.

If the system of linear equations $x + ay + z = 3$,$x + 2y + 2z = 6$,and $x + 5y + 3z = b$ has no solution,then: